Theorem sepnsepolem2 48839

Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 48840. Proof could be shortened by 1 step using ssdisjdr 48729. (Contributed by Zhi Wang, 1-Sep-2024.)

Hypothesis

Ref	Expression
sepnsepolem2.1	⊢ (𝜑 → 𝐽 ∈ Top)

Assertion

Ref	Expression
sepnsepolem2	⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥)   𝐽(𝑥)

Proof of Theorem sepnsepolem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sepnsepolem2.1	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		id 22	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 ∈ Top)
3		sslin 4214	. . . . 5 ⊢ (𝑧 ⊆ 𝑦 → (𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦))
4		sseq0 4374	. . . . . 6 ⊢ (((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑧) = ∅)
5	4	ex 412	. . . . 5 ⊢ ((𝑥 ∩ 𝑧) ⊆ (𝑥 ∩ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
6	3, 5	syl 17	. . . 4 ⊢ (𝑧 ⊆ 𝑦 → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
7	6	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑧 ⊆ 𝑦) → ((𝑥 ∩ 𝑦) = ∅ → (𝑥 ∩ 𝑧) = ∅))
8		ineq2 4185	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝑧))
9	8	eqeq1d 2732	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
10	9	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑧) = ∅))
11	2, 7, 10	opnneieqv 48827	. 2 ⊢ (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
12	1, 11	syl 17	1 ⊢ (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3055   ∩ cin 3921   ⊆ wss 3922  ∅c0 4304  ‘cfv 6519  Topctop 22786  neicnei 22990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-top 22787  df-nei 22991

This theorem is referenced by:  sepnsepo  48840